giving and receiving - Teageru, TeMorau, TeKureru

I already know how to use あげる、もらう and くれる as verbs... But I have some questions regarding the te form of a verb + てあげる、てもらう、てくれる. Where I'm learning it they translate Verb(teform)+あげる/もらう as making a favor like

友達がゲームをするのを教えてくれた。 My friend made me the favor of teaching me how to play the game. Is there a better way to translate it?

What different tone this sentence has in comparisson to something like:

友達がゲームをするのを教えた。?

Also, is くれる　only allowed when the action is going for me? When talking about a third person should I always use あげる or もらう?

"The mother took her children to the school."

母が彼女の子供に学校で連れてくれた. Is it wrong? How could it be right?

母が彼女の子供に学校で連れてもらった.

母が彼女の子供に学校で連れてあげた.

Also I can't really tell what different meaning the sentence would have changing the end like that, even though I know how to use it as verbs alone.

Sorry for the big question, I hope I am making sense.

Answer

• 母が子供を学校へ連れていった。
  
  
  
  • Mother took her children to school. (no favor)
  
  
  
  
  


• 子供が母に学校へ連れていってもらった。
  
  
  
  • The children were taken to school by mother. (favor to children by mom)
  
  
  
  


• 母が他人に（母の）子供を学校へ連れていってもらった。
  
  
  
  • Mother had another person take her children to school. (favor to mom by other)
  
  
  
  
  


• 母が他人の子供を学校へ連れていってあげた。
  
  
  
  • Mother took another persons children to school. (favor to the other by mom)
  
  
  
  

Favor here is not necessarily a really big generous favor.

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Edit in reply to comment:

When an outsider does something "for" your sister (an insider), then くれる　can also be used.

• Simple example: 他人が姉にドアを開けてくれた。


• Complex example: 他人が、姉に（姉の子供を学校へ連れていくことを）してくれた。

The complex sentence can be shortened to

• 他人が、姉の子供を学校へ連れていってくれた。(sentence accepted by one native speaker)

However, in that abbreviation, the favor-receiver is not specified. A friendly native speaker resource (not a linguist) feels uncomfortable with this sentence

• 他人が、姉に姉の子供を学校へ連れていってくれた。（sentence rejected by one native speaker)

Despite the fact that this would be OK

• 姉が、他人に姉の子供を学校へ連れていってもらった。(sentence accepted by one native speaker)

You used a complex example in your question, containing 連れていく. If you are only interested in the simple example, you can leave the complex example until later.